Civil Engineering Reference
In-Depth Information
∆
B
θ
B
Figure 4.15.
Example 4.6 Conjugate beam.
4.6
ViRtuAL WORK
Similar to the conjugate beam method, the virtual work method can be
used to obtain the displacement and slope at a specific point on a struc-
ture. The method was formulated by Gottfried Leibniz in 1695 (Leibniz
1695).
Virtual work
is based on internal strain energy from the member
and external work done by the forces. There are two ways to apply the
virtual work method. The first is to apply a virtual (unit) displacement
to find a real force. The second is to apply a virtual (unit) force to find a
real displacement. The force and the displacement in either case are in the
same direction. The basic equation to find a real displacement based on a
virtual force is as follows:
L
mM
EI
q
or ∆=
0
dx
(4.7)
The value of
M
is the moment equation due to the real loads on the struc-
ture. The value of
m
is the moment equation of the virtual load. Rotation or
deflection may be found depending on whether a virtual moment or force
is applied at the point under consideration.
Example 4.8
Virtual work
Determine the vertical deflection at the free end of the uniformly loaded,
cantilever beam in Figure 4.16 using the virtual work method.
A free-body diagram is drawn of the right-hand side of the beam to
determine the internal moment in the beam. The uniformly distributed
load is represented as a point load equal to the area under the load and
located at the centroid of the area.
From static equilibrium, we can determine the internal moment at any
point
x
measured from the right end of the beam.
M =−
wx
2
2
